Problem 20 - Precalculus: Mathematical for Calculus

Question

Use a graphing calculator to do the following. (a) Find the first 10 terms of the sequence. (b) Graph the first 10 terms of the sequence. $$a_{n}=4-2(-1)^{n}$$

Step-by-Step Solution

Answer

The sequence alternates between 6 and 2 for the first 10 terms. Graph shows oscillation between y=6 and y=2.

Step 1: Identify the sequence formula

The given sequence is defined by the formula \( a_{n} = 4 - 2(-1)^{n} \). This is a function of \( n \), where \( n \) is a positive integer that represents the term number in the sequence.

Step 2: Calculate first 10 terms

To find the first 10 terms, substitute values from 1 to 10 for \( n \) in the sequence formula and evaluate:- For \( n = 1 \), \( a_1 = 4 - 2(-1)^1 = 4 + 2 = 6 \)- For \( n = 2 \), \( a_2 = 4 - 2(-1)^2 = 4 - 2 = 2 \)- Continue this process up to \( n = 10 \). This gives us: \( a_1 = 6, a_2 = 2, a_3 = 6, a_4 = 2, a_5 = 6, a_6 = 2, a_7 = 6, a_8 = 2, a_9 = 6, a_{10} = 2 \).

Step 3: Graph the sequence terms

To graph the first 10 terms of the sequence, plot the terms on a coordinate plane with \( n \) on the x-axis and \( a_n \) on the y-axis.- (1, 6), (2, 2), (3, 6), (4, 2), (5, 6), (6, 2), (7, 6), (8, 2), (9, 6), (10, 2) are the points to be plotted. - Since the values alternate between 6 and 2, you will see a pattern of points oscillating between these two y-values.

Key Concepts

Understanding the Sequence Formula

A sequence formula is an expression that allows us to determine the terms of a sequence based on their position. In our given sequence, \( a_{n} = 4 - 2(-1)^{n} \), the formula depends on the variable \( n \), which represents the term number. The term numbers, also known as indices, begin from 1 onwards in this case.

To calculate a specific term, simply plug the term number into the formula in place of \( n \). For example, to find \( a_1 \), substitute \( n = 1 \) into the formula, yielding \( a_1 = 4 - 2(-1)^1 = 6 \). This process is repeated for each subsequent term.

The sequence formula acts like a blueprint, detailing how to construct each term based on its position in the sequence. Becoming comfortable with the algebraic manipulation of such formulas can greatly enhance one's ability to solve sequence-related problems.

Using a Graphing Calculator

Graphing calculators are a powerful tool in visualizing sequences. They can break down complex calculations and present data graphically, making it easier to understand patterns. When you input a sequence formula, such as \( a_{n} = 4 - 2(-1)^{n} \), into a graphing calculator, you can quickly obtain the sequence terms and their graphical representation.

To graph the sequence, enter the formula into the graphing calculator and specify the range of \( n \) values you are interested in (e.g., 1 to 10 for the first ten terms). The calculator will plot each point, showing \( n \) on the x-axis and \( a_n \) on the y-axis. This visual approach helps to identify patterns, such as alternating values, more intuitively than by simply calculating each term manually.

Using a graphing calculator efficiently comes with practice, but once mastered, it becomes a valuable aid in exploring more complex mathematical concepts and sequences.

Exploring Alternating Sequences

Alternating sequences are those where the terms switch between different values in a regular fashion. In our sequence, \( a_{n} = 4 - 2(-1)^{n} \), the terms alternate between 6 and 2. This occurs due to the \((-1)^n\) component, which changes sign with each increment of \( n \).

When \( n \) is odd, \((-1)^n\) equals -1, making \( 4 + 2 \), which gives a term of 6. Conversely, when \( n \) is even, \((-1)^n\) equals 1, so the term becomes \( 4 - 2 \), resulting in 2. This systematic sign flipping creates an alternating pattern.

Understanding such patterns is crucial for analyzing data sequences and can be observed in various applications, including alternating series in calculus or oscillating signals in engineering fields. Developing a keen eye for these patterns will enable more insight into both mathematical problems and real-world data.